"""
Node classes (`Apply`, `Variable`) and expression graph algorithms.
"""
from __future__ import print_function

from collections import deque
from copy import copy
from itertools import count

import theano
from theano import config
from theano.gof import utils
from six import string_types, integer_types, iteritems
from theano.misc.ordered_set import OrderedSet

__docformat__ = "restructuredtext en"

# Lazy imports to avoid circular dependencies.
is_same_graph_with_merge = None
equal_computations = None

NoParams = object()


class Node(utils.object2):
    """
    A Node in a theano graph.

    Graphs contain two kinds of Nodes -- Variable and Apply.
    Edges in the graph are not explicitly represented.
    Instead each Node keeps track of its parents via
    Variable.owner / Apply.inputs and its children
    via Variable.clients / Apply.outputs.

    """

    def get_parents(self):
        """
        Return a list of the parents of this node.
        Should return a copy--i.e., modifying the return
        value should not modify the graph structure.

        """
        raise NotImplementedError()


class Apply(Node):
    """
    An :term:`Apply` instance is a node in an expression graph which represents
    the application of an `Op` to some input `Variable` nodes, producing some
    output `Variable` nodes.

    This class is typically instantiated by an Op's make_node() function, which
    is typically called by that Op's __call__() function.

    An Apply instance serves as a simple structure with three important
    attributes:

    - :literal:`inputs` :  a list of `Variable` nodes that represent the
      arguments of the expression,

    - :literal:`outputs` : a list of `Variable` nodes that represent the
      variable of the expression, and

    - :literal:`op` : an `Op` instance that determines the nature of the
      expression being applied.

    The driver `compile.function` uses Apply's inputs attribute together with
    Variable's owner attribute to search the expression graph and determine
    which inputs are necessary to compute the function's outputs.

    A `Linker` uses the Apply instance's `op` field to compute the variables.

    Comparing with the Python language, an `Apply` instance is theano's version
    of a function call (or expression instance) whereas `Op` is theano's version
    of a function definition.

    Parameters
    ----------
    op : `Op` instance
    inputs : list of Variable instances
    outputs : list of Variable instances

    Notes
    -----
    The owner field of each output in the outputs list will be set to self.

    If an output element has an owner that is neither None nor self, then a
    ValueError exception will be raised.

    """

    def __init__(self, op, inputs, outputs):
        self.op = op
        self.inputs = []
        self.tag = utils.scratchpad()

        if not isinstance(inputs, (list, tuple)):
            raise TypeError("The inputs of an Apply must be a list or tuple")

        if not isinstance(outputs, (list, tuple)):
            raise TypeError("The output of an Apply must be a list or tuple")

        # filter inputs to make sure each element is a Variable
        for input in inputs:
            if isinstance(input, Variable):
                self.inputs.append(input)
            else:
                raise TypeError("The 'inputs' argument to Apply must contain Variable instances, not %s" % input)
        self.outputs = []
        # filter outputs to make sure each element is a Variable
        for i, output in enumerate(outputs):
            if isinstance(output, Variable):
                if output.owner is None:
                    output.owner = self
                    output.index = i
                elif output.owner is not self or output.index != i:
                    raise ValueError("All output variables passed to Apply must belong to it.")
                self.outputs.append(output)
            else:
                raise TypeError("The 'outputs' argument to Apply must contain Variable instances with no owner, not %s" % output)

    def run_params(self):
        """
        Returns the params for the node, or NoParams if no params is set.

        """
        if hasattr(self.op, 'get_params'):
            return self.op.get_params(self)
        return NoParams

    def __getstate__(self):
        d = self.__dict__
        # ufunc don't pickle/unpickle well
        if hasattr(self.tag, 'ufunc'):
            d = copy(self.__dict__)
            t = d["tag"]
            del t.ufunc
            d["tag"] = t
        return d

    def default_output(self):
        """
        Returns the default output for this node.

        Returns
        -------
        Variable instance
            An element of self.outputs, typically self.outputs[0].

        Notes
        -----
        May raise AttributeError self.op.default_output is out of range, or if
        there are multiple outputs and self.op.default_output does not exist.

        """
        do = getattr(self.op, 'default_output', None)
        if do is None:
            if len(self.outputs) == 1:
                return self.outputs[0]
            else:
                raise AttributeError(
                    "%s.default_output should be an output index." % self.op)
        elif not isinstance(do, integer_types):
            raise AttributeError("%s.default_output should be an int or long" %
                                 self.op)
        elif do < 0 or do >= len(self.outputs):
            raise AttributeError("%s.default_output is out of range." %
                                 self.op)
        return self.outputs[do]

    out = property(default_output,
                   doc="alias for self.default_output()")
    """
    Alias for self.default_output().

    """

    def __str__(self):
        return op_as_string(self.inputs, self)

    def __repr__(self):
        return str(self)

    def __asapply__(self):
        return self

    def clone(self):
        """
        Duplicate this Apply instance with inputs = self.inputs.

        Returns
        -------
        object
            A new Apply instance (or subclass instance) with new outputs.

        Notes
        -----
        Tags are copied from self to the returned instance.

        """
        cp = self.__class__(self.op, self.inputs,
                            [output.clone() for output in self.outputs])
        cp.tag = copy(self.tag)
        return cp

    def clone_with_new_inputs(self, inputs, strict=True):
        """
        Duplicate this Apply instance in a new graph.

        Parameters
        ----------
        inputs
            List of Variable instances to use as inputs.
        strict : bool
            If True, the type fields of all the inputs must be equal
            to the current ones (or compatible, for instance Tensor /
            CudaNdarray of the same dtype and broadcastable patterns,
            in which case they will be converted into current Type), and
            returned outputs are guaranteed to have the same types as
            self.outputs.  If False, then there's no guarantee that the
            clone's outputs will have the same types as self.outputs,
            and cloning may not even be possible (it depends on the Op).

        Returns
        -------
        object
            An Apply instance with the same op but different outputs.

        """
        assert isinstance(inputs, (list, tuple))
        remake_node = False
        new_inputs = inputs[:]
        for i, (curr, new) in enumerate(zip(self.inputs, new_inputs)):
            if not curr.type == new.type:
                if strict:
                    # If compatible, casts new into curr.type
                    new_inputs[i] = curr.type.filter_variable(new)
                else:
                    remake_node = True
        if remake_node:
            new_node = self.op.make_node(*new_inputs)
            new_node.tag = copy(self.tag).__update__(new_node.tag)
        else:
            new_node = self.clone()
            new_node.inputs = new_inputs
        return new_node

    def get_parents(self):
        return list(self.inputs)

    # convenience properties
    nin = property(lambda self: len(self.inputs), doc='same as len(self.inputs)')
    """
    Property: Number of inputs.

    """
    nout = property(lambda self: len(self.outputs), doc='same as len(self.outputs)')
    """
    Property: Number of outputs.

    """
    params_type = property(lambda self: self.op.params_type, doc='type to use for the params')


class Variable(Node):
    """
    A :term:`Variable` is a node in an expression graph that represents a
    variable.

    The inputs and outputs of every `Apply` (theano.gof.Apply) are `Variable`
    instances. The input and output arguments to create a `function` are also
    `Variable` instances. A `Variable` is like a strongly-typed variable in
    some other languages; each `Variable` contains a reference to a `Type`
    instance that defines the kind of value the `Variable` can take in a
    computation.

    A `Variable` is a container for four important attributes:

    - :literal:`type` a `Type` instance defining the kind of value this
      `Variable` can have,

    - :literal:`owner` either None (for graph roots) or the `Apply` instance
      of which `self` is an output,

    - :literal:`index` the integer such that :literal:`owner.outputs[index] is
      this_variable` (ignored if `owner` is None),

    - :literal:`name` a string to use in pretty-printing and debugging.

    There are a few kinds of Variables to be aware of: A Variable which is the
    output of a symbolic computation has a reference to the Apply instance to
    which it belongs (property: owner) and the position of itself in the owner's
    output list (property: index).

    - `Variable` (this base type) is typically the output of a symbolic
      computation.

    - `Constant` (a subclass) which adds a default and un-replaceable
      :literal:`value`, and requires that owner is None.

    - `TensorVariable` subclass of Variable that represents a numpy.ndarray
       object.

    - `TensorSharedVariable` Shared version of TensorVariable.

    - `SparseVariable` subclass of Variable that represents
      a scipy.sparse.{csc,csr}_matrix object.

    - `CudaNdarrayVariable` subclass of Variable that represents our object on
      the GPU that is a subset of numpy.ndarray.

    - `RandomVariable`.

    A Variable which is the output of a symbolic computation will have an owner
    not equal to None.

    Using the Variables' owner field and the Apply nodes' inputs fields, one can
    navigate a graph from an output all the way to the inputs. The opposite
    direction is not possible until a FunctionGraph has annotated the Variables
    with the clients field, ie, before the compilation process has begun a
    Variable does not know which Apply nodes take it as input.

    Parameters
    ----------
    type : a Type instance
        The type governs the kind of data that can be associated with this
        variable.
    owner : None or Apply instance
        The Apply instance which computes the value for this variable.
    index : None or int
        The position of this Variable in owner.outputs.
    name : None or str
        A string for pretty-printing and debugging.

    Examples
    --------

    .. code-block:: python

        import theano
        from theano import tensor

        a = tensor.constant(1.5)        # declare a symbolic constant
        b = tensor.fscalar()            # declare a symbolic floating-point scalar

        c = a + b                       # create a simple expression

        f = theano.function([b], [c])   # this works because a has a value associated with it already

        assert 4.0 == f(2.5)            # bind 2.5 to an internal copy of b and evaluate an internal c

        theano.function([a], [c])       # compilation error because b (required by c) is undefined

        theano.function([a,b], [c])     # compilation error because a is constant, it can't be an input

        d = tensor.value(1.5)           # create a value similar to the constant 'a'
        e = d + b
        theano.function([d,b], [e])     # this works.  d's default value of 1.5 is ignored.

    The python variables :literal:`a,b,c` all refer to instances of type
    `Variable`. The `Variable` refered to by `a` is also an instance of
    `Constant`.

    `compile.function` uses each `Apply` instance's `inputs` attribute together
    with each Variable's `owner` field to determine which inputs are necessary
    to compute the function's outputs.

    """

    # __slots__ = ['type', 'owner', 'index', 'name']
    __count__ = count(0)

    def __init__(self, type, owner=None, index=None, name=None):
        super(Variable, self).__init__()

        self.tag = utils.scratchpad()
        self.type = type
        if owner is not None and not isinstance(owner, Apply):
            raise TypeError("owner must be an Apply instance", owner)
        self.owner = owner
        if index is not None and not isinstance(index, integer_types):
            raise TypeError("index must be an int", index)
        self.index = index
        if name is not None and not isinstance(name, string_types):
            raise TypeError("name must be a string", name)
        self.name = name
        self.auto_name = 'auto_' + str(next(self.__count__))

    def __str__(self):
        """Return a str representation of the Variable.

        """
        if self.name is not None:
            return self.name
        if self.owner is not None:
            op = self.owner.op
            if self.index == op.default_output:
                return str(self.owner.op) + ".out"
            else:
                return str(self.owner.op) + "." + str(self.index)
        else:
            return "<%s>" % str(self.type)

    def __repr_test_value__(self):
        """Return a repr of the test value.

        Return a printable representation of the test value. It can be
        overridden by classes with non printable test_value to provide a
        suitable representation of the test_value.
        """
        return repr(theano.gof.op.get_test_value(self))

    def __repr__(self, firstPass=True):
        """Return a repr of the Variable.

        Return a printable name or description of the Variable. If
        config.print_test_value is True it will also print the test_value if
        any.
        """
        to_print = [str(self)]
        if config.print_test_value and firstPass:
            try:
                to_print.append(self.__repr_test_value__())
            except AttributeError:
                pass
        return '\n'.join(to_print)

    def clone(self):
        """
        Return a new Variable like self.

        Returns
        -------
        Variable instance
            A new Variable instance (or subclass instance) with no owner or
            index.

        Notes
        -----
        Tags are copied to the returned instance.

        Name is copied to the returned instance.

        """
        # return copy(self)
        cp = self.__class__(self.type, None, None, self.name)
        cp.tag = copy(self.tag)
        return cp

    def __lt__(self, other):
        raise NotImplementedError('Subclasses of Variable must provide __lt__',
                                  self.__class__.__name__)

    def __le__(self, other):
        raise NotImplementedError('Subclasses of Variable must provide __le__',
                                  self.__class__.__name__)

    def __gt__(self, other):
        raise NotImplementedError('Subclasses of Variable must provide __gt__',
                                  self.__class__.__name__)

    def __ge__(self, other):
        raise NotImplementedError('Subclasses of Variable must provide __ge__',
                                  self.__class__.__name__)

    def get_parents(self):
        if self.owner is not None:
            return [self.owner]
        return []

    def eval(self, inputs_to_values=None):
        """
        Evaluates this variable.

        Parameters
        ----------
        inputs_to_values
            A dictionary mapping theano Variables to values.

        Examples
        --------

        >>> import numpy
        >>> import theano.tensor as T
        >>> x = T.dscalar('x')
        >>> y = T.dscalar('y')
        >>> z = x + y
        >>> numpy.allclose(z.eval({x : 16.3, y : 12.1}), 28.4)
        True

        We passed :func:`eval` a dictionary mapping symbolic theano
        variables to the values to substitute for them, and it returned
        the numerical value of the expression.

        Notes
        -----

        `eval` will be slow the first time you call it on a variable --
        it needs to call :func:`function` to compile the expression behind
        the scenes. Subsequent calls to :func:`eval` on that same variable
        will be fast, because the variable caches the compiled function.

        This way of computing has more overhead than a normal Theano
        function, so don't use it too much in real scripts.
        """

        if inputs_to_values is None:
            inputs_to_values = {}

        if not hasattr(self, '_fn_cache'):
            self._fn_cache = dict()

        inputs = tuple(sorted(inputs_to_values.keys(), key=id))
        if inputs not in self._fn_cache:
            self._fn_cache[inputs] = theano.function(inputs, self)
        args = [inputs_to_values[param] for param in inputs]

        rval = self._fn_cache[inputs](*args)

        return rval

    def __getstate__(self):
        d = self.__dict__.copy()
        d.pop("_fn_cache", None)
        return d


class Constant(Variable):
    """
    A :term:`Constant` is a `Variable` with a `value` field that cannot be
    changed at runtime.

    Constant nodes make eligible numerous optimizations: constant inlining in
    C code, constant folding, etc.

    Notes
    -----
    The data field is filtered by what is provided in the constructor for the
    Constant's type field.

    WRITEME

    """

    # __slots__ = ['data']
    def __init__(self, type, data, name=None):
        Variable.__init__(self, type, None, None, name)
        self.data = type.filter(data)

    def equals(self, other):
        # this does what __eq__ should do, but Variable and Apply should always be hashable by id
        return isinstance(other, Constant) and self.signature() == other.signature()

    def signature(self):
        return (self.type, self.data)

    def merge_signature(self):
        return self.signature()

    def __str__(self):
        if self.name is not None:
            return self.name
        else:
            name = str(self.data)
            if len(name) > 20:
                name = name[:10] + '...' + name[-10:]
            return 'Constant{%s}' % name

    def clone(self):
        """
        We clone this object, but we don't clone the data to lower memory
        requirement. We suppose that the data will never change.

        """
        cp = self.__class__(self.type, self.data, self.name)
        cp.tag = copy(self.tag)
        return cp

    def __set_owner(self, value):
        """
        WRITEME

        Raises
        ------
        ValueError
            If `value` is not `None`.

        """
        if value is not None:
            raise ValueError("Constant instances cannot have an owner.")

    owner = property(lambda self: None, __set_owner)
    value = property(lambda self: self.data, doc='read-only data access method')

    # index is not defined, because the `owner` attribute must necessarily be None


def stack_search(start, expand, mode='bfs', build_inv=False):
    """
    Search through a graph, either breadth- or depth-first.

    Parameters
    ----------
    start : deque
        Search from these nodes.
    expand : callable
        When we get to a node, add expand(node) to the list of nodes to visit.
        This function should return a list, or None.

    Returns
    -------
    list of `Variable` or `Apply` instances (depends on `expend`)
        The list of nodes in order of traversal.

    Notes
    -----
    A node will appear at most once in the return value, even if it
    appears multiple times in the start parameter.

    :postcondition: every element of start is transferred to the returned list.
    :postcondition: start is empty.

    """

    if mode not in ('bfs', 'dfs'):
        raise ValueError('mode should be bfs or dfs', mode)
    rval_set = set()
    rval_list = list()
    if mode == 'bfs':
        start_pop = start.popleft
    else:
        start_pop = start.pop
    expand_inv = {}
    while start:
        l = start_pop()
        if id(l) not in rval_set:
            rval_list.append(l)
            rval_set.add(id(l))
            expand_l = expand(l)
            if expand_l:
                if build_inv:
                    for r in expand_l:
                        expand_inv.setdefault(r, []).append(l)
                start.extend(expand_l)
    assert len(rval_list) == len(rval_set)
    if build_inv:
        return rval_list, expand_inv
    return rval_list


def ancestors(variable_list, blockers=None):
    """
    Return the variables that contribute to those in variable_list (inclusive).

    Parameters
    ----------
    variable_list : list of `Variable` instances
        Output `Variable` instances from which to search backward through
        owners.

    Returns
    -------
    list of `Variable` instances
        All input nodes, in the order found by a left-recursive depth-first
        search started at the nodes in `variable_list`.

    """
    def expand(r):
        if r.owner and (not blockers or r not in blockers):
            return reversed(r.owner.inputs)
    dfs_variables = stack_search(deque(variable_list), expand, 'dfs')
    return dfs_variables


def inputs(variable_list, blockers=None):
    """
    Return the inputs required to compute the given Variables.

    Parameters
    ----------
    variable_list : list of `Variable` instances
        Output `Variable` instances from which to search backward through
        owners.

    Returns
    -------
    list of `Variable` instances
        Input nodes with no owner, in the order found by a left-recursive
        depth-first search started at the nodes in `variable_list`.

    """
    vlist = ancestors(variable_list, blockers)
    rval = [r for r in vlist if r.owner is None]
    return rval


def variables_and_orphans(i, o):
    """
    WRITEME

    """
    def expand(r):
        if r.owner and r not in i:
            l = list(r.owner.inputs) + list(r.owner.outputs)
            l.reverse()
            return l
    variables = stack_search(deque(o), expand, 'dfs')
    orphans = [r for r in variables if r.owner is None and r not in i]
    return variables, orphans


def ops(i, o):
    """
    WRITEME

    Parameters
    ----------
    i : list
        Input L{Variable}s.
    o : list
        Output L{Variable}s.

    Returns
    -------
    object
        The set of ops that are contained within the subgraph that lies
        between i and o, including the owners of the L{Variable}s in o and
        intermediary ops between i and o, but not the owners of the L{Variable}s
        in i.

    """
    ops = set()
    variables, orphans = variables_and_orphans(i, o)
    for r in variables:
        if r not in i and r not in orphans:
            if r.owner is not None:
                ops.add(r.owner)
    return ops


def variables(i, o):
    """
    WRITEME

    Parameters
    ----------
    i : list
        Input L{Variable}s.
    o : list
        Output L{Variable}s.

    Returns
    -------
    object
        The set of Variables that are involved in the subgraph that lies
        between i and o. This includes i, o, orphans(i, o) and all values of
        all intermediary steps from i to o.

    """
    return variables_and_orphans(i, o)[0]


def orphans(i, o):
    """
    WRITEME

    Parameters
    ----------
    i : list
        Input L{Variable}s.
    o : list
        Output L{Variable}s.

    Returns
    -------
    object
        The set of Variables which one or more Variables in o depend on but are
        neither in i nor in the subgraph that lies between i and o.

    Examples
    --------
    orphans([x], [(x+y).out]) => [y]

    """
    return variables_and_orphans(i, o)[1]


def clone(i, o, copy_inputs=True):
    """
    Copies the subgraph contained between i and o.

    Parameters
    ----------
    i : list
        Input L{Variable}s.
    o : list
        Output L{Variable}s.
    copy_inputs : bool
        If True, the inputs will be copied (defaults to True).

    Returns
    -------
    object
        The inputs and outputs of that copy.

    """
    equiv = clone_get_equiv(i, o, copy_inputs)
    return [equiv[input] for input in i], [equiv[output] for output in o]


def clone_get_equiv(inputs, outputs, copy_inputs_and_orphans=True, memo=None):
    """
    Return a dictionary that maps from Variable and Apply nodes in the
    original graph to a new node (a clone) in a new graph.

    This function works by recursively cloning inputs... rebuilding a directed
    graph from the bottom (inputs) up to eventually building new outputs.

    Parameters
    ----------
    inputs : a list of Variables
    outputs : a list of Variables
    copy_inputs_and_orphans : bool
        True means to create the cloned graph from new input and constant
        nodes (the bottom of a feed-upward graph).
        False means to clone a graph that is rooted at the original input
        nodes.
    memo : None or dict
        Optionally start with a partly-filled dictionary for the return value.
        If a dictionary is passed, this function will work in-place on that
        dictionary and return it.

    """
    if memo is None:
        memo = {}

    # clone the inputs if necessary
    for input in inputs:
        if copy_inputs_and_orphans:
            cpy = input.clone()
            cpy.owner = None
            cpy.index = None
            memo.setdefault(input, cpy)
        else:
            memo.setdefault(input, input)

    # go through the inputs -> outputs graph cloning as we go
    for apply in io_toposort(inputs, outputs):
        for input in apply.inputs:
            if input not in memo:
                if copy_inputs_and_orphans:
                    cpy = input.clone()
                    memo[input] = cpy
                else:
                    memo[input] = input

        new_apply = apply.clone_with_new_inputs([memo[i] for i in apply.inputs])
        memo.setdefault(apply, new_apply)
        for output, new_output in zip(apply.outputs, new_apply.outputs):
            memo.setdefault(output, new_output)

    # finish up by cloning any remaining outputs (it can happen)
    for output in outputs:
        if output not in memo:
            memo[output] = output.clone()

    return memo


def general_toposort(r_out, deps, debug_print=False,
                     compute_deps_cache=None, deps_cache=None,
                     clients=None):
    """
    WRITEME

    Parameters
    ----------
    deps
        A python function that takes a node as input and returns its dependence.
    compute_deps_cache : optional
        If provided deps_cache should also be provided. This is a function like
        deps, but that also cache its results in a dict passed as deps_cache.
    deps_cache : dict
        Must be used with compute_deps_cache.
    clients : dict
        If a dict is passed it will be filled with a mapping of node
        -> clients for each node in the subgraph.

    Notes
    -----
        deps(i) should behave like a pure function (no funny business with
        internal state).

        deps(i) will be cached by this function (to be fast).

        The order of the return value list is determined by the order of nodes
        returned by the deps() function.

        deps should be provided or can be None and the caller provides
        compute_deps_cache and deps_cache. The second option removes a Python
        function call, and allows for more specialized code, so it can be
        faster.

    """
    if compute_deps_cache is None:
        deps_cache = {}

        def compute_deps_cache(io):
            if io not in deps_cache:
                d = deps(io)
                if d:
                    if not isinstance(d, (list, OrderedSet)):
                        raise TypeError(
                            "Non-deterministic collections here make"
                            " toposort non-deterministic.")
                    deps_cache[io] = list(d)
                else:
                    deps_cache[io] = d
                return d
            else:
                return deps_cache[io]
    assert deps_cache is not None

    assert isinstance(r_out, (tuple, list, deque))

    reachable, _clients = stack_search(deque(r_out), compute_deps_cache,
                                       'dfs', True)
    if clients is not None:
        clients.update(_clients)
    sources = deque([r for r in reachable if not deps_cache.get(r, None)])

    rset = set()
    rlist = []
    while sources:
        node = sources.popleft()
        if node not in rset:
            rlist.append(node)
            rset.add(node)
            for client in _clients.get(node, []):
                deps_cache[client] = [a for a in deps_cache[client]
                                      if a is not node]
                if not deps_cache[client]:
                    sources.append(client)

    if len(rlist) != len(reachable):
        if debug_print:
            print('')
            print(reachable)
            print(rlist)
        raise ValueError('graph contains cycles')

    return rlist


def io_toposort(inputs, outputs, orderings=None, clients=None):
    """
    WRITEME

    Parameters
    ----------
    inputs : list or tuple of Variable instances
    outputs : list or tuple of Apply instances
    orderings : dict
        Key: Apply instance. Value: list of Apply instance.
        It is important that the value be a container with a deterministic
        iteration order. No sets allowed!
    clients : dict
        If a dict is provided it will be filled with mappings of
        node->clients for each node in the subgraph that is sorted

    """
    # the inputs are used only here in the function that decides what 'predecessors' to explore
    iset = set(inputs)

    # We build 2 functions as a speed up
    deps_cache = {}

    compute_deps = None
    compute_deps_cache = None
    if not orderings:  # can be None or empty dict
        # Specialized function that is faster when no ordering.
        # Also include the cache in the function itself for speed up.
        def compute_deps_cache(obj):
            if obj in deps_cache:
                return deps_cache[obj]
            rval = []
            if obj not in iset:
                if isinstance(obj, Variable):
                    if obj.owner:
                        rval = [obj.owner]
                elif isinstance(obj, Apply):
                    rval = list(obj.inputs)
                if rval:
                    if not isinstance(rval, (list, OrderedSet)):
                        raise TypeError(
                            "Non-deterministic collections here make"
                            " toposort non-deterministic.")
                    deps_cache[obj] = list(rval)
                else:
                    deps_cache[obj] = rval
            else:
                deps_cache[obj] = rval
            return rval
    else:
        def compute_deps(obj):
            rval = []
            if obj not in iset:
                if isinstance(obj, Variable):
                    if obj.owner:
                        rval = [obj.owner]
                elif isinstance(obj, Apply):
                    rval = list(obj.inputs)
                rval.extend(orderings.get(obj, []))
            else:
                assert not orderings.get(obj, [])
            return rval

    topo = general_toposort(outputs, deps=compute_deps,
                            compute_deps_cache=compute_deps_cache,
                            deps_cache=deps_cache, clients=clients)
    return [o for o in topo if isinstance(o, Apply)]


default_leaf_formatter = str


def default_node_formatter(op, argstrings):
    return "%s(%s)" % (op.op, ", ".join(argstrings))


def io_connection_pattern(inputs, outputs):
    """
    Returns the connection pattern of a subgraph defined by given
    inputs and outputs.

    """
    inner_nodes = io_toposort(inputs, outputs)

    # Initialize 'connect_pattern_by_var' by establishing each input as
    # connected only to itself
    connect_pattern_by_var = {}
    nb_inputs = len(inputs)

    for i in range(nb_inputs):
        input = inputs[i]
        inp_connection_pattern = [i == j for j in range(nb_inputs)]
        connect_pattern_by_var[input] = inp_connection_pattern

    # Iterate through the nodes used to produce the outputs from the
    # inputs and, for every node, infer their connection pattern to
    # every input from the connection patterns of their parents.
    for n in inner_nodes:

        # Get the connection pattern of the inner node's op. If the op
        # does not define a connection_pattern method, assume that
        # every node output is connected to every node input
        try:
            op_connection_pattern = n.op.connection_pattern(n)
        except AttributeError:
            op_connection_pattern = ([[True] * len(n.outputs)] *
                                     len(n.inputs))

        # For every output of the inner node, figure out which inputs it
        # is connected to by combining the connection pattern of the inner
        # node and the connection patterns of the inner node's inputs.
        for out_idx in range(len(n.outputs)):
            out = n.outputs[out_idx]
            out_connection_pattern = [False] * nb_inputs

            for inp_idx in range(len(n.inputs)):
                inp = n.inputs[inp_idx]

                if inp in connect_pattern_by_var:
                    inp_connection_pattern = connect_pattern_by_var[inp]

                    # If the node output is connected to the node input, it
                    # means it is connected to every inner input that the
                    # node inputs is connected to
                    if op_connection_pattern[inp_idx][out_idx]:
                        out_connection_pattern = [out_connection_pattern[i] or
                                                  inp_connection_pattern[i]
                                                  for i in range(nb_inputs)]

            # Store the connection pattern of the node output
            connect_pattern_by_var[out] = out_connection_pattern

    # Obtain the global connection pattern by combining the
    # connnection patterns of the individual outputs
    global_connection_pattern = [[] for o in range(len(inputs))]
    for out in outputs:
        out_connection_pattern = connect_pattern_by_var[out]
        for i in range(len(inputs)):
            global_connection_pattern[i].append(out_connection_pattern[i])

    return global_connection_pattern


def is_same_graph(var1, var2, givens=None, debug=False):
    """
    Return True iff Variables `var1` and `var2` perform the same computation.

    By 'performing the same computation', we mean that they must share the same
    graph, so that for instance this function will return False when comparing
    (x * (y * z)) with ((x * y) * z).

    The current implementation is not efficient since, when possible, it
    verifies equality by calling two different functions that are expected to
    return the same output. The goal is to verify this assumption, to
    eventually get rid of one of them in the future.

    Parameters
    ----------
    var1
        The first Variable to compare.
    var2
        The second Variable to compare.
    givens
        Similar to the `givens` argument of `theano.function`, it can be used
        to perform substitutions in the computational graph of `var1` and
        `var2`. This argument is associated to neither `var1` nor `var2`:
        substitutions may affect both graphs if the substituted variable
        is present in both.
    debug : bool
        If True, then an exception is raised when we are in a situation where
        the `equal_computations` implementation cannot be called.
        This parameter is intended to be used in tests only, to make sure we
        properly test both implementations.

    Examples
    --------

        ======  ======  ======  ======
        var1    var2    givens  output
        ======  ======  ======  ======
        x + 1   x + 1   {}      True
        x + 1   y + 1   {}      False
        x + 1   y + 1   {x: y}  True
        ======  ======  ======  ======

    """
    # Lazy import.
    if givens is None:
        givens = {}
    global equal_computations, is_same_graph_with_merge
    if equal_computations is None:
        from theano.gof.opt import is_same_graph_with_merge
        from theano.scan_module.scan_utils import equal_computations
    # Convert `givens` to dictionary.
    if not isinstance(givens, dict):
        givens = dict(givens)
    # Get result from the merge-based function.
    rval1 = is_same_graph_with_merge(var1=var1, var2=var2, givens=givens)
    # Get result from the function `equal_computations` from scan_utils.

    use_equal_computations = True
    if givens:
        # We need to build the `in_xs` and `in_ys` lists. To do this, we need
        # to be able to tell whether a variable belongs to the computational
        # graph of `var1` or `var2`.
        # The typical case we want to handle is when `to_replace` belongs to
        # one of these graphs, and `replace_by` belongs to the other one. In
        # other situations, the current implementation of `equal_computations`
        # is probably not appropriate, so we do not call it.
        ok = True
        in_xs = []
        in_ys = []
        # Compute the sets of all variables found in each computational graph.
        inputs_var = list(map(inputs, ([var1], [var2])))
        all_vars = [set(variables(v_i, v_o))
                    for v_i, v_o in ((inputs_var[0], [var1]),
                                     (inputs_var[1], [var2]))]

        def in_var(x, k):
            # Return True iff `x` is in computation graph of variable `vark`.
            return x in all_vars[k - 1]

        for to_replace, replace_by in iteritems(givens):
            # Map a substitution variable to the computational graphs it
            # belongs to.
            inside = dict((v, [in_var(v, k) for k in (1, 2)])
                          for v in (to_replace, replace_by))
            if (inside[to_replace][0] and not inside[to_replace][1] and
                    inside[replace_by][1] and not inside[replace_by][0]):
                # Substitute variable in `var1` by one from `var2`.
                in_xs.append(to_replace)
                in_ys.append(replace_by)
            elif (inside[to_replace][1] and not inside[to_replace][0] and
                  inside[replace_by][0] and not inside[replace_by][1]):
                # Substitute variable in `var2` by one from `var1`.
                in_xs.append(replace_by)
                in_ys.append(to_replace)
            else:
                ok = False
                break
        if not ok:
            # We cannot directly use `equal_computations`.
            if debug:
                raise AssertionError(
                    'When `debug` is True we want to make sure we are also '
                    'using the `equal_computations` implementation')
            use_equal_computations = False
    else:
        in_xs = None
        in_ys = None
    if use_equal_computations:
        rval2 = equal_computations(xs=[var1], ys=[var2],
                                   in_xs=in_xs, in_ys=in_ys)
        assert rval2 == rval1
    return rval1


def op_as_string(i, op,
                 leaf_formatter=default_leaf_formatter,
                 node_formatter=default_node_formatter):
    """
    WRITEME

    """
    strs = as_string(i, op.inputs, leaf_formatter, node_formatter)
    return node_formatter(op, strs)


def as_string(i, o,
              leaf_formatter=default_leaf_formatter,
              node_formatter=default_node_formatter):
    """
    WRITEME

    Parameters
    ----------
    i : list
        Input `Variable` s.
    o : list
        Output `Variable` s.
    leaf_formatter : theano.function
        Takes a `Variable`  and returns a string to describe it.
    node_formatter : callable
        Takes an `Op`  and the list of strings corresponding to its arguments
        and returns a string to describe it.

    Returns
    -------
    str
        Returns a string representation of the subgraph between i and o. If the
        same op is used by several other ops, the first occurrence will be
        marked as :literal:`*n -> description` and all subsequent occurrences
        will be marked as :literal:`*n`, where n is an id number (ids are
        attributed in an unspecified order and only exist for viewing
        convenience).

    """
    i = set(i)

    orph = orphans(i, o)

    multi = set()
    seen = set()
    for output in o:
        op = output.owner
        if op in seen:
            multi.add(op)
        else:
            seen.add(op)
    for op in ops(i, o):
        for input in op.inputs:
            op2 = input.owner
            if input in i or input in orph or op2 is None:
                continue
            if op2 in seen:
                multi.add(op2)
            else:
                seen.add(input.owner)
    multi = [x for x in multi]
    done = set()

    def multi_index(x):
        return multi.index(x) + 1

    def describe(r):
        if r.owner is not None and r not in i and r not in orph:
            op = r.owner
            idx = op.outputs.index(r)
            if len(op.outputs) == 1:
                idxs = ""
            else:
                idxs = "::%i" % idx
            if op in done:
                return "*%i%s" % (multi_index(op), idxs)
            else:
                done.add(op)
                s = node_formatter(op, [describe(input) for input in op.inputs])
                if op in multi:
                    return "*%i -> %s" % (multi_index(op), s)
                else:
                    return s
        else:
            return leaf_formatter(r)

    return [describe(output) for output in o]


def view_roots(r):
    """
    Utility function that returns the leaves of a search through
    consecutive view_map()s.

    WRITEME

    """
    owner = r.owner
    if owner is not None:
        try:
            view_map = owner.op.view_map
            view_map = dict((owner.outputs[o], i)
                            for o, i in iteritems(view_map))
        except AttributeError:
            return [r]
        if r in view_map:
            answer = []
            for i in view_map[r]:
                answer += view_roots(owner.inputs[i])
            return answer
        else:
            return [r]
    else:
        return [r]


def list_of_nodes(inputs, outputs):
    """
    Return the apply nodes of the graph between inputs and outputs.

    """
    return stack_search(
        deque([o.owner for o in outputs]),
        lambda o: [inp.owner for inp in o.inputs
                   if inp.owner and
                   not any(i in inp.owner.outputs for i in inputs)])
